BASIC MATRIX OPERATIONS Mr. Velazquez

Honors Precalculus

MATRIX NOTATIONWe can represent a matrix in two ways:

1. An uppercase letter such as A, B or C.

2. A lowercase letter enclosed in brackets.

A general element in matrix 𝐴 is denoted

by 𝑎𝑖𝑗 , which refers to the element in the

𝑖th row and 𝑗th column.

e.g. The element 𝑎32 is located in the 3rd row, 2nd

column of matrix 𝐴

A matrix of order 𝑚 × 𝑛 has 𝑚 rows and

𝑛 columns.

If 𝑚 = 𝑛, the matrix has the same number

of rows as columns, and we would call this

a square matrix.

23 31

1 3 3

Let A= 4 5 3

3 2 4

. What is the order of A?

b. If A= , identify a , and ai j

a

a

− − − − − −

EQUALITY OF MATRICES

MATRIX ADDITION AND SUBTRACTION

MATRIX ADDITION AND SUBTRACTION

0 0Zero matrix =

0 0

3 2 3 2Additive Inverses: The inverse of is

1 5 1 5

3 2 3 2 0 0

1 5 1 5 0 0

If A

− −

− −

− − + =

− −

3 2 3 2= then -A =

1 5 1 5

− −

− −

MATRIX ADDITION AND SUBTRACTION

Perform the given matrix operations:

MATRIX SCALAR MULTIPLICATION

MATRIX SCALAR MULTIPLICATION

MATRIX SCALAR MULTIPLICATION

MATRIX MULTIPLICATIONThere are a few rules worth remembering when

multiplying matrices:

1. We can only multiply two matrices (𝐴 × 𝐵)

if the number of columns of the first matrix

𝐴 is equal to the number of rows of the second matrix 𝐵. Otherwise, the result is

undefined.

2. If the order of 𝐴 is 𝑚 × 𝑛 and the order of

𝐵 is 𝑛 × 𝑝, then the resulting matrix will be

of order 𝑚 × 𝑝.

3. Matrix multiplication is NOT commutative,

meaning that 𝐴 × 𝐵 is likely to give a different result than 𝐵 × 𝐴

MATRIX MULTIPLICATION

MATRIX MULTIPLICATION

MATRIX MULTIPLICATION

MATRIX MULTIPLICATION

Multiply the following matrices:

MATRIX MULTIPLICATIONMultiply the following matrices:

3 −15 −2 21 0 3

5 −2 21 0 3

3−1

(a) (b)

APPLICATIONS OF MATRICESAll of the images that you see on the Web have been created or manipulated on a computer in a digital format made up of hundreds of thousands or even millions of tiny squares called pixels. Pixels are created by dividing an image into a grid. The computer can change the brightness of every square or pixel in this grid. See the letter L created in the illustration below using color levels of pixels. Black has color level 3, dark gray 2, on down to white which is 0. See the matrix that we get with these color and placement designations.

APPLICATIONS OF MATRICESUsing the color level grid below, define a 5 × 5 matrix that displays the lower case letter a.

CLASSWORK & HOMEWORKCLASSWORK: MATRIX OPERATIONS

Using the methods previously outlined and the color levels below,

define two matrices that would display your initials in a pixel grid.

HOMEWORK:

Khan Academy

Due 5/3